In many communication systems, including both wired and wireless transmission systems, there are strict limitations on transmit signal bandwidth. Such limitations impose a demand for signal modulation with a number of levels greater than two. Many conventional systems employ Trellis-coded modulation (TCM) in such applications.
There is a growing demand for communication systems, including both wired and emerging wireless transmission systems, that require modulation to be accomplished with a number of levels greater than two, mainly due to strict limitations on transmit signal bandwidth. Trellis-coded modulation (TCM) is an example of a conventional modulation scheme for such applications. However, a problem associated with TCM is that it is unsuitable for iterative decoding. Therefore, further improvements in signal quality at an acceptable complexity are difficult to achieve.
“A turbo TCM scheme with low decoding complexity, ” Catena Networks Inc., Temporary Document BI-090, ITU-T Study Group 15, Question 4, Goa, India, 23–27 Oct. 2000, “Proposal of decision making for turbo coding and report of performance evaluation of proposed TTCM(PCCC) with R-S code and without R-S code,” Mitsubishi Electric Corp., Temporary Document BI-003, ITU-T Study Group 15, Goa, India, 23–27 Oct. 2000, and “Results of the requirements requested in the coding ad hoc report,” Vocal Technologies Inc., Temporary Document HC-073, ITU-T Study Group 15, Question 4, Huntsville, Canada, 31 Jul.–4 Aug. 2000, describe turbo-coding schemes for multilevel ADSL and VDSL transmission. These turbo-coding techniques involve encoding of the information bits by parallel concatenation of convolutional encoders in recursive systematic form and iterative decoding by one of several possible turbo-decoding techniques. “Block product turbo codes for G.dmt.bis and G.lite.bis.”Globespan Inc., Temporary Document BA-063, ITU-T Study Group 15, Question 4, Antwerp, Belgium, 19–23 Jun. 2000 describes the application of block product codes using component Bose-Chaudhuri-Hoequenghem (BCH) codes and their soft iterative decoding based on the Chase algorithm. These techniques offer some performance enhancements over Trellis coding at the expense of incurring additional complexity.
Another coding technique uses Low Density Parity Check (LDPC) block codes. As indicated in R. G. Gallager, “Low-density parity-check codes,” IRE Trans. Info. Theory, vol. IT-8, pp. 21–28. January 1962, D. J. C. MacKay and R. M. Neal, “Near Shannon limit performance of low density parity check codes, Electron. Lett. vol. 32, no. 18, pp. 1645–1646, August 1996, D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices.” IEEE Trans. on Inform. Theory, vol. 45, No. 2, pp. 399–431, March 1999, and FOSSORIER, M. P. C., MIHALJEVIC, M., and IMAI, H.: “Reduced complexity iterative decoding of low density parity check codes based on belief propagation”, IEEE Trans. Commun., 1999, 47, (5), pp. 673–680, coded modulation using LDPC codes has to date focussed on applications requiring binary modulation such as wireless systems or digital magnetic recording.
K. R. Narayanan and J. Li, “Bandwidth efficient low density parity check coding using multilevel coding and iterative multistage decoding,” Proc. Int. Symp. on Turbo-Codes, Brest, France, pp. 165–168, September 2000 describes a multilevel coding technique based on binary LDPC block codes. This technique uses LDPC codes for bit-interleaved modulation or for multilevel coding with iterative multi-stage decoding. For bit-interleaved LDPC modulation according to this technique, all the bits used to select a multilevel symbol are LDPC code bits. For multilevel coding, several LDPC codes are used as component codes in a multilevel scheme. This technique has the drawback of requiring more than one LDPC encoder/decoder, leading to substantial implementation complexity especially for long codes and/or large constellation sizes.
“Low density parity check coded modulation for ADSL,” Aware Inc., Temporary Document BI-081, ITU-T Study Group 15, Question 4, Goa, India, 23–27 Oct. 2000 also describes a multilevel coding technique based on binary LDPC block codes. This technique is similar to TCM, except that LDPC coding is employed instead of convolutional coding. In particular, set partitioning follows the same principle as that used in TCM. This technique has the drawback of requiring an additional Bose-Chaudhuri-Hoeguenghem (BCH) code which adds to system complexity. Also, set partitioning, as required in TCM and similar schemes, leads to poor performance for soft-decision based decoding techniques.
LDPC codes can be decoded via the sum-product algorithm (SPA). The SPA is described in the aforementioned reference D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. on Inform. Theory, vol. 45, No. 2, pp. 399–431, March 1999. The SPA operates on a bipartite graph associated with a given sparse parity check matrix H having M rows and N columns. This graph has two types of nodes: N symbol nodes corresponding to each bit in a code word x, and M check nodes corresponding to the parity checks pcm(x), 1≦m≦M represented by the rows of the matrix H. Each symbol node is connected to the check nodes it participates in, and each check node is connected to the symbol nodes it checks. The SPA operates by passing messages between symbol nodes and check nodes. The messages themselves can be a posteroiri probabilities (APP) or log likelihood ratios (LLRs). Typical message parsing schedules alternately compute updates of all symbol nodes and of all check nodes.
The computational complexity of the SPA is governed by the check node updates. In the probability domain, such computation involves the summation of product terms each involving a plurality of probabilities. In the log domain, the check node updates require computation of the inverse hyperbolic tangent of a product of hyperbolic tangent functions of LLRs. Mackay demonstrated via computational simulations that there is a loss in performance of approximately 0.2 dB associated with such conventional techniques. This performance loss can be substantial in terms of block and symbol error rates because of the steepness of the error curves of LDPC codes. A decoding algorithm having a substantially reduced complexity but without incurring a loss in performance would be clearly desirable.